Archive for the ‘2012’ Category

The list version of the Borodin-Kostochka Conjecture for graphs with large maximum degree

Ilkyoo Choi (최일규)
Department of Mathematics, University of Illinois at Urbana-Champaign, IL, USA

2012/12/27 Thu 4PM-5PM

Brooks’ Theorem states that for a graph G with maximum degree Δ(G) at least 3, the chromatic number is at most Δ(G) when the clique number is at most Δ(G). Vizing proved that the list chromatic number is also at most Δ(G) under the same conditions. Borodin and Kostochka conjectured that a graph G with maximum degree at least 9 must be (Δ(G)-1)-colorable when the clique number is at most Δ(G)-1; this was proven for graphs with maximum degree at least 1014 by Reed. We prove an analogous result for the list chromatic number; namely, we prove that a graph G with Δ(G)≥ 1020 is (Δ(G)-1)-choosable when the clique number is at most Δ(G)-1. This is joint work with H. A. Kierstead, L. Rabern, and B. Reed.

In this talk, the diagrams of affine permutations and their balanced labellings will be introduced. As in the finite case, which was investigated by Fomin, Greene, Reiner, and Shimozono, the balanced labellings give a natural encoding of reduced decompositions of affine permutations. In fact, we show that the sum of weight monomials of the column strict balanced labellings is the affine Stanley symmetric function defined by Lam. The affine Stanley symmetric function is the object of active research in the field of Schubert calculus. It is the affine counterpart of the Stanley symmetric function which is the limit of Schubert polynomials. Our construction is a natural tableau-theoretic realization of this function. We also give a simple algorithm to recover reduced words from balanced labellings. Applying this theory, we will give a necessary and sufficient condition for a diagram to be an affine permutation diagram. If time allows, we will introduce some conjectures about when the affine Stanley symmetric functions coincide. This talk is based on the joint work with Taedong Yun.

The Matroid intersection theorem of Edmonds gives a formula for the maximum size of a common independent set in two matroids on the same ground set. Aharoni and Berger generalized this theorem to the `topological’ setting where one of the matroids is replaced by an arbitrary simplicial complex. I will present two applications of this result to graph-theoretical problems. The first application is related to the existence of spanning 2-walks in tough graphs, the other one is more recent and gives a bound on the fractional arboricity of a graph G ensuring that G can be covered by k forests and a matching. In both cases, slightly better results can be obtained by other methods, but there seems to be room for improvement on the topological side as well.

Graphs, tournaments, colouring and containment

Paul Seymour

Some tournaments H are heroes; they have the property that all tournaments not containing H as a subtournament have bounded chromatic number (colouring a tournament means partitioning its vertex-set into transitive subsets). In joint work with eight authors, we found all heroes explicitly. That was great fun, and it would be nice to find an analogue for graphs instead of tournaments.
The problem is too trivial for graphs, if we only exclude one graph H; but it becomes fun again if we exclude a finite set of graphs. The Gyarfas-Sumner conjecture says that if we exclude a forest and a clique then chromatic number is bounded. So what other combinations of excluded subgraphs will give bounded chromatic (or cochromatic) number? It turns out (assuming the Gyarfas-Sumner conjecture) that for any finite set S of graphs, the graphs not containing any member of S all have bounded cochromatic number if and only if S contains a complete multipartite graph, the complement of a complete multipartite graph, a forest, and the complement of a forest.
Proving this led us to the following: for every complete multipartite graph H, and every disjoint union of cliques J, there is a number n with the following property. For every graph G, if G contains neither of H,J as an induced subgraph, then V(G) can be partitioned into two sets such that the first contains no n-vertex clique and the second no n-vertex stable set.
In turn, this led us (with Alex Scott) to the following stronger result. Let H be the disjoint union of H_1,H_2, and let J be obtained from the disjoint union of J_1,J_2 by making every vertex of J_1 adjacent to every vertex of J_2. Then there is a number n such that for every graph G containing neither of H,J as an induced subgraph, V(G) can be partitioned into n sets such that for each of them, say X, one of H_1,H_2,J_1,J_2 is not contained in G|X.
How about a tournament analogue of this? It exists, and the same (short) proof works; and this leads to a short proof of the most difficult result of the heroes paper that we started with.
There are a number of other related results and open questions. Joint work with Maria Chudnovsky.

Excluding paths and antipaths

Maria Chudnovsky

The Erdos-Hajnal conjecture states that for every graph H, there exists a constant delta(H)>0, such that every n-vertex graph with no induced subgraph isomorphic to H contains a clique or a stable set of size at least n^delta(H). This conjecture is still open. We consider a variant of the conjecture, where instead of excluding a single graph H as an induced subgraph, a family of graphs is excluded. We prove this modified conjecture for the case when the five-edge path and its complement are excluded. Our second result is an asymmetric version of this: we prove that for every graph G such that G contains no induced six-edge path, and the complement of G contains no induced four-edge path, G contains a polynomial-size clique or stable set. This is joint work with Paul Seymour.

Quasirandomness and property testing of permutations

Daniel Kráľ

A systematic study of large combinatorial objects has recently led to discovering many connections between discrete mathematics and analysis. In this talk, we explore the analytic view of large permutations. We associate every sequence of permutations with a measure on a unit square and show the following: if the density of every 4-element subpermutations in a permutation p is 1/4!+o(1), then the density of every k-element subpermutation is 1/k!+o(1). This solves a question of Graham whether quasirandomness of a permutation is captured by densities of its 4-element subpermutations. At the end of the talk, we present a result related to an area of computer science called property testing. A property tester is an algorithm which determines (with a small error probability) properties of a large input object based on a small sample of it. Specifically, we prove a conjecture of Hoppen, Kohayakawa, Moreira and Sampaio asserting that hereditary properties of permutations are testatble with respect to the so-called Kendal’s tau distance.
The results in this talk were obtained jointly with Tereza Klimosova or Oleg Pikhurko.

I’ll explain how to construct, given a set system, an inclusion-exclusion formula valid for that set system and with size sensitive to the size of the Venn diagram of the set system. The main ingredient of the proof are random simplicial complexes. The talk will start from first principle.
This is joint work with J. Matousek, M. Tancer, Z. Safernova and P. Patak from Charles university.

For a hypergraph G and a positive integer c, let χ&ell;(G,c) be the minimum value of &ell; such that G is L-colorable from every list L with |L(v)|=&ell; for each v&in;V(G) and |L(u)∩L(v)|≤c for all e=uv&in;E(G). This parameter was studied by Kratochvíl, Tuza and Voigt for various kinds of graphs. In this talk, we present the asymptotics of χ&ell;(G,c) for complete graphs, complete multipartite graphs and hypergraphs. This is a joint work with Z. Füredi and A. Kostochka.